Remark 1.4 This corollary can also be deduced from [1]} by constructing invariant measures carried by sub-subshifts on which the IFSs satisfy OSC that approach the box-counting dimension of these image sets from below. The slightly new thing here is that, as Theorem REF  applies to all invariant measures, the above dimension is actually reached by two invariant measures carried on \(X\)  and \(Y\)  respectively.

